DIOPHANTINE CONDITIONS IMPLY CRITICAL-POINTS ON THE BOUNDARIES OF SIEGEL DISKS OF POLYNOMIALS

Let f be a polynomial map of the Riemann sphere of degree at least two , We prove that if f has: a Siegel disk G on which the rotation number satisfies a diophantine condition, then either the boundary B of G co ntains a critical point or B is a Lakes of Wada indecomposable continu um with one of the lakes containing a critical point, Consequently, if the boundary B of G has only 2 complementary domains, then B contains a critical point. We also show, without any assumption on the rotatio n number, that each proper nondegenerate subcontinuum of the boundary B of G is tree-like, and any other bounded complementary domain of B i s a preperiodic component of the grand orbit of G. Finally, we establi sh some conditions under which B contains no periodic point.